A ray of light of intensity \(I\) is incident on a parallel glass slab at point \(A\) as shown in diagram. It undergoes partial reflection and refraction. At each reflection, \(25\%\) of incident energy is reflected. The rays \(AB\) and \(A'B'\) undergo interference. The ratio of \(I_{max}\) and \(I_{min}\) is

Three infinitely long charged thin sheets are placed as shown in figure. The magnitude of electric field at the point \(P\) is \(\frac{x \sigma}{\epsilon_0}\). The value of \(x\) is _______. (all quantities are measured in \(SI\) units).

Given below are two statements :
Statement-I : The equivalent emf of two nonideal batteries connected in parallel is smaller than either of the two emfs.
Statement-II : The equivalent internal resistance of two nonideal batteries connected in parallel is smaller than the internal resistance of either of the two batteries.
In the light of the above statements, choose the correct answer from the options given below.

A black coloured solid sphere of radius \(R\) and mass \(M\) is inside a cavity with vacuum inside. The walls of the cavity are maintained at temperature \(T_0\). The initial temperature of the sphere is \(3T_0\). If the specific heat of the material of the sphere varies as \(\alpha T^3\) per unit mass with the temperature \(T\) of the sphere, where \(\alpha \) is a constant, then the time taken for the sphere to cool down to temperature \(2T_0\) will be ( \(\sigma \) is Stefan Boltzmann constant)

The width of one of the two slits in Young's double slit experiment is d while that of the other slit is \(x \mathrm{~d}\). If the ratio of the maximum to the minimum intensity in the interference pattern on the screen is \(9: 4\) then what is the value of \(x\) ?
(Assume that the field strength varies according to the slit width.)

Diameter of a plano-convex lens is \(6 \,cm\) and thickness at the centre is \(3\, mm\). If the speed of light in the material of the lens is \(2  \times  10^8 m/sec\), the focal length of the lens is.......\(cm\)

A ray of light passing through an equilateral prism is having velocity \(2.12 \times 10^8\) m/s in the prism material, then the minimum angle of deviation is _________ degrees.

Let \(N\) be the sum of the numbers appeared when two fair dice are rolled and let the probability that \(N -2, \sqrt{3 N }, N +2\) are in geometric progression be \(\frac{ k }{48}\). Then the value of \(k\) is

A circle passes through the points \((2, 3)\) and \((4, 5)\). If its centre lies on the line, \(y- 4x + 3 = 0\) , then its radius is equal to

Let \(\vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{d}}=\vec{a} \times \overrightarrow{\mathrm{b}}\). If \(\overrightarrow{\mathrm{c}}\) is a vector such that \(\vec{a} \cdot \overrightarrow{\mathrm{c}}=|\overrightarrow{\mathrm{c}}|\), \(|\overrightarrow{\mathrm{c}}-2 \vec{a}|^2=8\) and the angle between \(\overrightarrow{\mathrm{d}}\) and \(\overrightarrow{\mathrm{c}}\) is \(\frac{\pi}{4}\), then \(|10-3 \overrightarrow{\mathrm{~b}} \cdot \overrightarrow{\mathrm{c}}|+|\overrightarrow{\mathrm{d}} \times \overrightarrow{\mathrm{c}}|^2\) is equal to

If you are provided a set of resistances \(2\, \Omega, 4\, \Omega\) \(6\, \Omega\) and \(8\, \Omega\). Connect these resistances so as to obtain an equivalent resistance of \(\frac{46}{3}\, \Omega\).

Let \(S\) be the sum of all solutions (in radians) of the equation \(\sin ^{4} \theta+\cos ^{4} \theta-\sin \theta \cos \theta=0\) in \([0,4 \pi]\) Then \(\frac{8 \mathrm{~S}}{\pi}\) is equal to ...... .

Given below two statements:
Statement I: \(25^{13}+20^{13}+8^{13}+3^{13}\) is divisible by 7.
Statement II: The integral part of \((7+4\sqrt{3})^{25}\) is an odd number.
In the light of the above statements, choose the correct answer from the options given below